A008466 -id:A008466 - cp's OEIS frontend

A351529 The number of quaternary strings of length n containing 00.

0, 0, 1, 7, 40, 205, 991, 4612, 20905, 92935, 407056, 1762117, 7556095, 32148940, 135892321, 571232647, 2389810360, 9956870845, 41335010911, 171055514452, 705891052825, 2905717608775, 11934337612576, 48918212175157, 200149835407615, 817572886925980

Offset: 0

R. J. Mathar, Feb 13 2022

Index entries for linear recurrences with constant coefficients, signature (7,-9,-12).

Cf. A008466 (2-ary), A186244 (3-ary), A351530 (5-ary), A125145 (not containing 00).

LinearRecurrence[{7,-9,-12},{0,0,1},30] (* Harvey P. Dale, Feb 27 2023 *)

G.f.: x^2 / ( (4*x-1)*(3*x^2+3*x-1) ).

a(n) = 4^n - A125145(n).

A353501 Number of integer partitions of n with all parts and all multiplicities > 2.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 2, 3, 0, 0, 6, 2, 0, 6, 3, 2, 9, 2, 5, 11, 3, 5, 18, 6, 4, 20, 13, 8, 26, 10, 17, 37, 14, 16, 51, 23, 24, 58, 38, 32, 75, 44, 52, 100, 52, 59, 143, 75, 77, 159, 114, 112, 203, 132, 154, 266, 175

Offset: 0

Gus Wiseman, May 16 2022

nonn

			The a(n) partitions for selected n (A = 10):
  n=9:   n=12:   n=21:      n=24:       n=30:
------------------------------------------------------
  (333)  (444)   (777)      (888)       (AAA)
         (3333)  (444333)   (6666)      (66666)
                 (3333333)  (444444)    (555555)
                            (555333)    (666444)
                            (4443333)   (777333)
                            (33333333)  (6663333)
                                        (55533333)
                                        (444333333)
                                        (3333333333)

The version for only parts > 2 is A008483.
The version for only multiplicities > 2 is A100405.
The version for parts and multiplicities > 1 is A339222, ranked by A062739.
For prime parts and multiplicities we have A351982, compositions A353429.
The version for compositions is A353428 (partial A078012, A353400).
These partitions are ranked by A353502.
A000726 counts partitions with all mults <= 2, compositions A128695.
A004250 counts partitions with some part > 2, compositions A008466.
A137200 counts compositions with all parts and run-lengths <= 2.
Cf. A003242, A047966, A055923, A098859, A114901, A325534, A333755, A335464.

Table[Length[Select[IntegerPartitions[n],Min@@#>2&&Min@@Length/@Split[#]>2&]],{n,0,30}]

A354235 Heinz numbers of integer partitions with at least one part divisible by 3.

Original entry on oeis.org

5, 10, 13, 15, 20, 23, 25, 26, 30, 35, 37, 39, 40, 45, 46, 47, 50, 52, 55, 60, 61, 65, 69, 70, 73, 74, 75, 78, 80, 85, 89, 90, 91, 92, 94, 95, 100, 103, 104, 105, 110, 111, 113, 115, 117, 120, 122, 125, 130, 135, 137, 138, 140, 141, 143, 145, 146, 148, 150

Offset: 1

Gus Wiseman, May 23 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
    5: {3}
   10: {1,3}
   13: {6}
   15: {2,3}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   26: {1,6}
   30: {1,2,3}
   35: {3,4}
   37: {12}
   39: {2,6}
   40: {1,1,1,3}
   45: {2,2,3}
   46: {1,9}
   47: {15}
   50: {1,3,3}
   52: {1,1,6}
   55: {3,5}
   60: {1,1,2,3}

For 4 instead of 3 we have A046101, counted by A295342.
This sequence ranks the partitions counted by A295341, compositions A335464.
For 2 instead of 3 we have A324929 (and A013929), counted by A047967.
A001222 counts prime factors with multiplicity, distinct A001221.
A004250 counts partitions with some part > 2, compositions A008466.
A004709 lists numbers divisible by no cube, counted by A000726.
A036966 lists 3-full numbers, counted by A100405.
A046099 lists non-cubefree numbers.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A354234 counts partitions of n with at least one part divisible by k.
Cf. A000720, A003963, A008483, A018256, A062739, A064428, A117485, A181819, A339222, A353508.

Select[Range[100],MemberQ[PrimePi/@First/@If[#==1,{},FactorInteger[#]]/3,_?IntegerQ]&]

A119587 2^n + 1 - 2*Fibonacci(n+1).

Original entry on oeis.org

0, 1, 1, 3, 7, 17, 39, 87, 189, 403, 847, 1761, 3631, 7439, 15165, 30795, 62343, 125905, 253783, 510759, 1026685, 2061731, 4136991, 8295873, 16627167, 33311647, 66716029, 133582107, 267406999, 535206833, 1071049287

Offset: 0

Ross La Haye, May 31 2006, Jun 27 2007

nonn

Index entries for linear recurrences with constant coefficients, signature (4,-4,-1,2).

Cf. A000032, A000045, A000051, A000071, A006355, A008466, A027934, A101220, A118654.

Table[2^n + 1 - 2 Fibonacci[n + 1], {n, 0, 30}]

a(n) = 2^n + 1 - 2*Fibonacci(n+1) = 2^n + 1 + Fibonacci(n) - Fibonacci(n+3) = 2^n + 1 - Fibonacci(n) - Lucas(n). a(n) = 2(2^(n-1) - Fibonacci(n+1)) + 1, for n > 0. a(n) = A000051(n) - A006355(n+2) = A000051(n) - A000045(n) - A000032(n). a(n) = A101220(2,2,n-1) - A101220(1,1,n-3), for n > 2. a(n) = A008466(n) - A000071(n-1), for n > 0. a(n) = 2*A008466(n-1) + 1, for n > 0.
a(n) = 2*A101220(2,2,n-2) + 1, for n > 1. a(n) = Sum[2^(n-k)Fibonacci(k) - Fibonacci(k-2),{k,0,n}] = antidiagonal sums of A118654. a(n+1) - a(n) = 2(2^(n-1) - Fibonacci(n)), for n > 0. a(n+1) - a(n) = 2*A027934(n-2), for n > 1. a(n+1) - a(n) = 2*A101220(1,2,n-1), for n > 0. a(0) = 0; a(1) = 1; a(n) = a(n-1) + a(n-2) + 2^(n-2) - 1, for n > 1. a(0) = 0; a(1) = 1; a(2) = 1; a(3) = 3; a(n) = 4*a(n-1) - 4*a(n-2) - a(n-3) + 2*a(n-4), for n > 3.
O.g.f. = x(1-3x+3x^2)/((1-x)(1-2x)(1-x-x^2)).

A153281 Triangle read by rows, A130321 * A127647. Also, number of subsets of [n+2] with consecutive integers that start at k.

Original entry on oeis.org

1, 2, 1, 4, 2, 2, 8, 4, 4, 3, 16, 8, 8, 6, 5, 32, 16, 16, 12, 10, 8, 64, 32, 32, 24, 20, 16, 13, 128, 64, 64, 48, 40, 32, 26, 21, 256, 128, 128, 96, 80, 64, 52, 42, 34, 512, 256, 256, 192, 160, 128, 104, 84, 68, 55

Offset: 0

Gary W. Adamson, Dec 23 2008

Row sums = A008466(k-2): (1, 3, 8, 19, 43, 94, ...).

T(n,k) is the number of subsets of {1,...,n+2} that contain consecutive integers and that have k as the first integer in the first consecutive string. (See the example below.) Hence rows sums of T(n,k) give the number of subsets of {1,...,n+2} that contain consecutive integers. Also, T(n,k) = F(k)*2^(n+1-k), where F(k) is the k-th Fibonacci number, since there are F(k) subsets of {1,...,k-2} that contain no consecutive integers and there are 2^(n+1-k) subsets of {k+2,...,n+2}. [Dennis P. Walsh, Dec 21 2011]

			First few rows of the triangle:
    1;
    2,   1;
    4,   2,   2;
    8,   4,   4,   3;
   16,   8,   8,   6,   5;
   32,  16,  16,  12,  10,   8;
   64,  32,  32,  24,  20,  16,  13;
  128,  64,  64,  48,  40,  32,  26,  21;
  256, 128, 128,  96,  80,  64,  52,  42,  34;
  512, 256, 256, 192, 160, 128, 104,  84,  68,  55;
  ...
Row 4 = (16, 8, 8, 6, 5) = termwise products of (16, 8, 4, 2, 1) and (1, 1, 2, 3, 5).
For n=5 and k=3, T(5,3)=16 since there are 16 subsets of {1,2,3,4,5,6,7} containing consecutive integers with 3 as the first integer in the first consecutive string, namely,
  {1,3,4}, {1,3,4,5}, {1,3,4,6}, {1,3,4,7}, {1,3,4,5,6}, {1,3,4,5,7}, {1,3,4,6,7}, {1,3,4,5,6,7}, {3,4}, {3,4,5}, {3,4,6}, {3,4,7}, {3,4,5,6}, {3,4,5,7}, {3,4,6,7}, and {3,4,5,6,7}. [_Dennis P. Walsh_, Dec 21 2011]

Cf. A008466, A127647, A130321.

with(combinat, fibonacci):
seq(seq(2^(n+1-k)*fibonacci(k),k=1..(n+1)),n=0..10);

Table[2^(n+1-k) Fibonacci[k],{n,0,10},{k,n+1}]//Flatten (* Harvey P. Dale, Apr 26 2020 *)

Triangle read by rows, A130321 * A127647. A130321 = an infinite lower triangular matrix with powers of 2: (A000079) in every column: (1, 2, 4, 8, ...).
A127647 = an infinite lower triangular matrix with the Fibonacci numbers, A000045 as the main diagonal and the rest zeros.
T(n,k)=2^(n+1-k)*F(k) where F(k) is the k-th Fibonacci number. [Dennis Walsh, Dec 21 2011]

A167826 a(n) is the number of n-tosses having a run of 3 or more heads and a run of 3 or more tails for a fair coin.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 8, 26, 74, 194, 482, 1152, 2674, 6068, 13524, 29704, 64460, 138482, 294988, 623834, 1311086, 2740666, 5702270, 11815752, 24395678, 50209572, 103048168, 210965064, 430938832, 878534170

Offset: 1

V.J. Pohjola, Nov 13 2009

G. C. Greubel, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (4,-3,-3,0,3,2).

Cf. A000045, A000073.

Cf. A008466, A050231, A167821.

b[1] = 0; b[2] = 1; b[3] = 1; b[n_]: = b[n-1] + b[n-2] + b[n-3]; Table[2^n - 2*(Sum[b[n + 3 - i], {i, 1, 3}] - Fibonacci[n + 1]), {n, 1, 30}]
LinearRecurrence[{4, -3, -3, 0, 3, 2}, {0, 0, 0, 0, 0, 2}, 50] (* G. C. Greubel, Jun 27 2016 *)

a(n) = 2^n - 2*(tribonacci(n+3) - Fibonacci(n+1)), where tribonacci = A000073.
From R. J. Mathar, Feb 06 2010: (Start)
a(n) = 4*a(n-1) - 3*a(n-2) - 3*a(n-3) + 3*a(n-5) + 2*a(n-6).
G.f.: -2*x^6/((2*x-1)*(x^2+x-1)*(x^3+x^2+x-1)). (End)

A272099 Triangle read by rows, T(n,k) = C(n+1,k+1)*F([k-n, k-n-1], [-n-1], -1), where F is the generalized hypergeometric function, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 4, 1, 12, 5, 1, 32, 18, 6, 1, 80, 56, 25, 7, 1, 192, 160, 88, 33, 8, 1, 448, 432, 280, 129, 42, 9, 1, 1024, 1120, 832, 450, 180, 52, 10, 1, 2304, 2816, 2352, 1452, 681, 242, 63, 11, 1, 5120, 6912, 6400, 4424, 2364, 985, 316, 75, 12, 1

Offset: 0

Peter Luschny, Apr 25 2016

This triangle results when the first column is removed from A210038. - Georg Fischer, Jul 26 2023

			Triangle starts:
1;
4,    1;
12,   5,    1;
32,   18,   6,   1;
80,   56,   25,  7,   1;
192,  160,  88,  33,  8,   1;
448,  432,  280, 129, 42,  9,  1;
1024, 1120, 832, 450, 180, 52, 10, 1;

A258109 (row sums), A008466 (alternating row sums), A001787 (col. 0), A001793 (col. 1), A055585 (col. 2).

Cf. A210038.

T := (n,k) -> binomial(n+1,k+1)*hypergeom([k-n, k-n-1], [-n-1], -1):
seq(seq(simplify(T(n,k)),k=0..n),n=0..9);

T[n_, k_] := Binomial[n+1, k+1] HypergeometricPFQ[{k-n, k-n-1}, {-n-1}, -1];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 22 2019 *)

cp's OEIS Frontend

A351529 The number of quaternary strings of length n containing 00.

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A353501 Number of integer partitions of n with all parts and all multiplicities > 2.

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A354235 Heinz numbers of integer partitions with at least one part divisible by 3.

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A119587 2^n + 1 - 2*Fibonacci(n+1).

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A153281 Triangle read by rows, A130321 * A127647. Also, number of subsets of [n+2] with consecutive integers that start at k.

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A167826 a(n) is the number of n-tosses having a run of 3 or more heads and a run of 3 or more tails for a fair coin.

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A272099 Triangle read by rows, T(n,k) = C(n+1,k+1)*F([k-n, k-n-1], [-n-1], -1), where F is the generalized hypergeometric function, for n>=0 and 0<=k<=n.

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